For each function, a) determine whether it is one-to-one; b)...

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Key Concepts

Monotonicity

Monotonicity refers to functions that are either entirely non-decreasing or non-increasing over their domains. A monotonic function is inherently one-to-one, which makes it easier to determine whether an inverse function exists. Recognizing that a function is monotonic can simplify analysis, as it directly implies injectivity.

Inverse Functions

An inverse function reverses the effect of the original function by swapping the roles of the independent and dependent variables. To find an inverse, you typically replace y with f(x), swap x and y in the equation, and then solve for y. The process relies on the original function being one-to-one so that the inverse mapping is well-defined.

One-to-One Functions

A function is considered one-to-one (or injective) if every output is produced by a unique input; that is, no two different inputs produce the same output. This property is essential when determining if an inverse function exists, as only one-to-one functions have inverses that are also functions.

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